476 COMPUTER-ASSISTED METHOD DEVELOPMENT emphasize commercial computer-simulation software that can predict separation as a function of one or more experimental conditions, by means of experimental data from a few preliminary separations. Because of this software’s ability to predict separation—as opposed to carrying out ‘‘real’’ experiments—we can reduce the amount of experimental work that is required while ensuring the ‘‘best’’ conditions for the final method. A simple example for isocratic RPC is the prediction of separation as a func- tion of mobile-phase strength (%B); for this application, two experimental runs are required prior to computer simulation. In the example of Figure 10.1, exper- iments using a mobile phase of 10 and 20% B (other conditions unchanged) are shown in Figure 10.1a,b. Resulting data are entered into the computer: retention times for each solute in each run, %B values for the two runs (10, 20% B), and other experimental conditions (A- and B-solvent compositions, column dimen- sions and particle size, flow rate, temperature; see Section 10.2.1). The computer can now be interrogated for predictions of separation (simulations) as a func- tion of changes in mobile-phase %B, flow rate, column dimensions, and particle size. The most useful information provided by computer simulation is usually a resolution map, as illustrated in Figure 10.1c for the separation of this mixture of seven acids and bases. A resolution map is a plot of the critical resolution R s , for the two least-resolved peaks, as a function of the condition or conditions that were varied in the initial experimental runs (Section 6.3.3 and Fig. 6.13b provide additional information on resolution maps). In this example %B was varied, so R s is plotted as a function of %B. In Figure 10.1c we see that maximum resolution occurs for three different conditions: 9, 17, and 25% B, with the largest resolution for 17% B. We can also request a simulated chromatogram for any value of %B, as illustrated in Figure 10.1d–f for the latter ‘‘optimum’’ %B values. The computer can provide further information for each of these (and other) separations as a function of %B; the range in values of k is especially useful in this regard (indicated for each of the separations in Figs. 10.1d–f ). For the usually recommended range of 1 ≤ k ≤ 10, it is seen that the separation of Figure 10.1f (25% B, 1 ≤ k ≤ 6) comes closest to this goal. However, some laboratories might prefer the separation of 17% B (3 ≤ k ≤ 12) in Figure 10.1e for its greater resolution. For the present sample, however, the retention range for any one of these three separations might be regarded as acceptable, depending on the goals of separation. Once an acceptable %B has been selected, it may be possible to trade excess resolution for a shorter run time, by changing column conditions. For example, the resolution of the separation with 25% B (R s = 2.4) could be reduced without adverse effects by a decrease in column length and/or an increase in flow rate. Computer simulation allows the user to explore the effects of changing column dimensions, particle size, or flow rate. Figure 10.1g shows one such simulation (for 25% B): a reduction in column length by half (from 150 to 75 mm), a change in particle size from5to3μm, and no change in flow rate. Acceptable resolution (R s = 2.1) and pressure (P = 2200 psi) are achieved in a run time of only 3 minutes, while retaining a preferred retention range of 1 ≥ k ≥ 6. The final choice of an ‘‘optimized’’ separation may depend on considerations other than resolution, run time, and retention range; for example, an acceptable 10.1 INTRODUCTION 477 0 Time (min) 0 Time (min) 5 1 6 3 7 4 2 5 1 2 3 + 6 4 7 10% B 20% B (a) (b) (c) R s 3.0 2.0 1.0 0.0 9% B9% B 17% B 25% B 10 20 30 %B (d) (e) 020 Time (min) 9% B 8 ≤ k ≤ 23 25% B 1 ≤ k ≤ 6 17% B 3 ≤ k ≤ 12 0246810 Time (min) 5 1 6 2 3 7 4 5 1 2 6 3 4 7 (f ) (g) 0246 Time (min) 5 1 2 3 6 4 7 0 Time (min) 25% B, 75-mm, 3-μm column R s = 2.1 213 Figure 10.1 Illustration of computer simulation for the separation of a mixture of acids and bases. Sample (a mixture of acids and bases) and conditions are the same as in Figure 7.7. (a,b) Initial experimental separations; (c) the resolution map predicted from the data of (a, b); (d–f) predicted chromatograms for various values of %B (corresponding to maximum resolution); (g) predicted chromatogram for conditions that favor fast separation (with adequate resolu- tion). Computer simulations based on data of [1]. column pressure, narrow peaks for more sensitive detection, and the relative impor- tance of different peaks in the chromatogram. For this reason, computer-simulation software should be flexible, so as to allow the user to examine the consequences of various changes in separation conditions—rather than just offer a single, ‘‘opti- mized’’ separation to the user. 478 COMPUTER-ASSISTED METHOD DEVELOPMENT 10.1.1 Basis and History of Computer Simulation Computer simulation makes use of various empirical and theoretical relationships; for example, Equations (2.7), (2.10), (2.18), (2.23), and (2.26), and Equations (9.7) to (9.14a), as means for predicting retention time and peak width for either isocratic or gradient elution. Often computer simulation begins with a relationship between values of k and %B (or φ = 0.01 %B). For the case of RPC, logk = logk w − Sφ (10.1) Here k w is the value of k for φ = 0 (i.e., water as mobile phase), and S is a constant for a given solute when all conditions except %B are held fixed. Two actual separations, where either %B (for isocratic elution) or gradient time (for gradient elution; Section 9.2.4.2) is varied, permit the calculations of values of k w and S for each solute in the sample. For NPC a similar relationship between k and φ exists (Eq. 8.4a). Once values of k w and S are known, values of k and retention time t R can then be predicted for any values of %B (or gradient time), column dimensions, or flow rate (Eqs. 2.5 and 9.10). Peak widths W can be calculated in various ways: (1) the assumption of some value of the plate number N for the initial separations (e.g., N = 10,000 for a 100-mm column with 3-μm particles), (2) measurements of peak widths for the initial experimental runs, or (3) calculation of peak widths based on values of N from Equation (2.17). Computer simulation can be extended to the case of segmented gradients, columns of different size, changes in flow rate, and so forth, by various relationships [2]. The accuracy of computer simulation has been confirmed in numerous experimental studies, as summarized in [2]. Retention times are usually reliable within a few percent while, more important, predicted values of resolution R s are typically accurate to ±10%, which is generally adequate for use in method development. The first example of computer simulation for HPLC was reported in 1978 by Laub and Purnell [3]; they described the use of a resolution map for isocratic RPC as a function of temperature. A few years later [4], Glajch, Kirkland, et al., reported an isocratic procedure for optimizing solvent type, based on the experimental plan of Figure 10.2f , which requires seven experimental runs with different proportions of ACN, MeOH, and THF (see also Section 6.4.1.1; Fig. 6.24). At the same time [5], Deming and coworkers presented a similar scheme for simultaneously optimizing mobile-phase pH and the concentration of an ion-pair reagent. The isocratic approach of Glajch and Kirkland was extended to gradient elution in 1983 [6]. In 1985 DryLab R software was introduced and subsequently expanded into the most comprehensive and widely used computer-simulation software presently available [7, 8]. The latter software is described in Section 10.2, as an example of various possible applications of computer simulation. Similar computer-simulation programs were developed by others after 1985, as reviewed and/or compared in [9–12] and summarized in Section 10.3.4. 10.1.2 When to Use Computer Simulation Method development can be pursued with or without the help of computer simula- tion, so it is important to weigh the potential pros and cons of computer simulation for each application. 10.1 INTRODUCTION 479 50°C 35% B 30°C 35% B 50°C 45% B 30°C 45% B (a) 50°C t G = 10 min 30°C t G = 10 min 30°C t G = 30 min 50°C t G = 30 min (b) pH−2.5 t G = 10 min pH−3.0 t G = 10 min pH−3.5 t G = 10 min pH−2.5 t G = 30 min pH−3.0 t G = 30 min pH−3.5 t G = 30 min (d) ACN MeOH 1:1 THF 1:1 1:1 1:1:1 (f ) 30% ACN 15% ACN+ 20% MeOH 40% MeOH 40% ACN 20% ACN+ 25% MeOH 50% MeOH (e) pH−2.5 35% B pH−3.0 35% B pH−3.5 35% B pH−2.5 45% B pH−3.0 45% B pH−3.5 45% B (c) Figure 10.2 Examples of different systematic approaches for optimizing separation selectivity (experimental design); (a, c, e, f) isocratic experiments, (b, d)gradients. 10.1.2.1 Advantages The value of computer simulation is likely to be greatest when one or more of the following conditions applies: • separations by gradient elution • complex samples that contain 5 to 10 components or more • very short run times are required (e.g., a minute or less) • method robustness is critical • minor improvements in resolution or run time are worthwhile The isocratic separation of a sample that contains only a few components—and where run time is not critical—may be developed quickly and easily by means of a 480 COMPUTER-ASSISTED METHOD DEVELOPMENT few trial-and-error experiments. In such cases the advantage of computer simulation is often marginal, although this overlooks other possible benefits of computer simulation (ability to fine-tune separation for faster separation, explore method robustness, etc.). Gradient elution, on the other hand, involves additional separation variables that are more conveniently explored via computer simulation. Similarly complex samples with lots of closely bunched peaks often require a large number of experiments to achieve a final successful separation; many of these experiments can be conveniently replaced by computer simulation. Very short run times can present a similar challenge, one also requiring several experiments. When method robustness is critical, computer simulation allows a quick examination of the effects of changes in different conditions on the separation, without a need for additional experiments. Small improvements in separation via trial-and-error changes in conditions can be effected easily by means of computer simulation. Simultaneous variation in two separation conditions that affect selectivity (Table 2.2) is a powerful tool that is often required for complex samples or other demanding separations. In such cases computer simulation can be used to determine optimal conditions by means of a minimum number of experimental runs (Fig. 10.2). Changes in relative retention as conditions are varied can confuse peak identification and the interpretation of experimental runs, especially when a large number of components are present in the sample. With computer simulation individual peaks are automatically matched between runs (following peak tracking for the initial experiments); the interpretation of simulated experiments is thereby greatly simplified. Examples of each of these various advantages of computer simulation will be shown in Section 10.2. 10.1.2.2 Disadvantages The cost of computer simulation software, and the time required for the chromatog- rapher to become familiar with its usage, represent the main impediments to a wider use of this approach. When method development is carried out infrequently, and samples are usually easy to separate, the potential value of computer simulation may be at best marginal. Another barrier to the use of computer simulation is the belief that the chromatographer has no need for computer simulation, because of his or her experience and competence. While computer simulation is seldom essential for carrying out method development, it can frequently reduce cost and improve the quality of the final method, even for experienced chromatographers. On the other hand, computer simulation is not a substitute for competence; chromatographic skills are still important for its effective use. A less important objection to computer simulation is that the predicted chromatograms are ‘‘ideal’’ rather than ‘‘real.’’ Thus baselines are assumed not to drift, peaks may be assumed to be symmetrical (although peak tailing can be taken into account by computer simulation; Section 10.2.3.5), and baseline artifacts or extraneous peaks are usually ignored. However, such chromatographic artifacts often detract from a final method, and are best eliminated before computer simulation is started. In other cases artifacts may not affect the interpretation of the separation or the choice of final conditions, and can be ignored. Finally, any mistakes in data entry (including mismatched peaks) can result in major errors in predicted separations. However, errors in data entry can be reduced 10.2 COMPUTER-SIMULATION SOFTWARE 481 by the automatic transfer of data from the data system to the computer-simulation program. Other errors that might occur from the use of computer simulation are usu- ally obvious—and easily corrected—when predicted separations and (confirming) experimental chromatograms do not agree. 10.2 COMPUTER-SIMULATION SOFTWARE Computer simulation is best used within an overall strategy of method development, as described in other sections of this book and illustrated in Section 10.4. Thus initial simulations should first examine the ‘‘best’’ retention range (values of k for isocratic elution or k ∗ for gradient elution) by varying isocratic %B or gradient time. At the same time any changes in selectivity as a function of %B or t G should be noted; maximum resolution may correspond to an intermediate value of (as in Fig. 10.1c). If further changes in relative retention are needed, other variables that affect selectivity should be explored next. The various plans (‘‘experimental designs’’) of Figure 10.2 summarize some of the more popular approaches. After one of these sets of experiments is carried out (e.g., four actual runs in the plan of Fig. 10.2a), the effects of simultaneous changes of two different separation conditions can be simulated. In principle, any two separation conditions that affect selectivity can be modeled as in Figure 10.2. After experiments have been carried out according to any of the choices of Figure 10.2, computer simulation can be used to select the best isocratic or gradient conditions (Section 9.2.2); column conditions (column dimensions, particle size, flow rate) can also be varied as a means of increasing resolution or decreasing run time. 10.2.1 DryLab Operation This section illustrates some useful features of computer simulation that form part of the DryLab software. Many of these features can also be found in other commercial computer-simulation software. An experimental design and separation mode are first selected, which defines the number of experimental calibration runs that will be required for computer simulation (as in Fig. 10.2). In this chapter we will limit our discussion to computer simulation for RPC. We will select the experimental design of Figure 10.2b as example, for the isocratic separation of a mixture of eight corticosteroids—but based on initial gradient-elution experiments. This requires four initial runs at two different values of gradient time t G (20, 60 min) and temperature T (30, 60 ◦ C), in order to simultaneously optimize %B and temperature T. The following experimental data are entered into the computer (see Fig. 10.3a): equipment dwell volume (5.5 mL; used only for initial experiments by gradient elution), column dimensions (250 × 4.6 mm), particle size (5 μm), flow rate (2.0 mL/min), mobile phase composition (‘‘elution data;’’ water, A; acetonitrile, B), gradient range (0–100% B), and retention times plus peak areas for each compound in each of the four experimental runs. Finally, it is necessary to carry out peak matching (Section 10.2.4), where retention data for each peak are matched with a given compound in the sample. Peak matching can be carried out manually, or facilitated by the computer; the peaks of Figure 10.3a 482 COMPUTER-ASSISTED METHOD DEVELOPMENT (a) (b) 38% B, 30°C; 25-cm, 3-μm column; 1.0 mL/min R s = 2.1; P = 2960 psi 020 Time (min) 10 (c) Figure 10.3 Examples of data entry (a) (gradient data) and laboratory (b) screens for iso- cratic computer simulation by means of DryLab. The sample is a mixture of eight corticos- teroids [13]. Conditions in (b): 250 × 4.6-mm C 18 column (5-μm particles); mobile phase, acetonitrile-water; 38% B; 30 ◦ C; 2.0 mL/min; (c) computer simulation for separation of sample with conditions of (b), except for a change in particle size (3 μm) and flow rate (1.0 mL/min). have been matched satisfactorily (as confirmed by the ‘‘area dev’’ column, which measures the %-deviation of predicted from actual peak areas). With the completion of data entry, a Laboratory screen for computer simula- tions can be selected (Fig. 10.3b). Since predictions for an isocratic separation are desired, the %B- ◦ C output mode is selected. Several display options are shown in the tabs above the resolution map: resolution map, resolution table, results table, and so forth. A resolution map has been selected in the example of Figure 10.3b; 10.2 COMPUTER-SIMULATION SOFTWARE 483 values of R s are indicated in shades of gray for various combinations of ◦ C(y-axis) and %B (x-axis). In this case maximum R s = 1.34 occurs for T = 30 ◦ Cand%B = 38% (note the cross-hairs in Fig. 10.3b). When the cursor is placed within the resolution map on the values of %B and T for maximum resolution (or any selected set of values of %B and T), a chromatogram for those conditions is immediately displayed (bottom of Figure 10.3b). An alternative choice of conditions (32.6% B, 30 ◦ C, when entered into the ‘‘status’’ box) provides a similar resolution (R s = 1.30), but requires a longer run time (22 min vs.12 min for the example of Fig. 10.3b). Further considerations that mitigate against the selection of T = 30 ◦ C and 32.6% B are as follows: First, the retention range is less favorable (4 ≤ k ≤ 16, compared with 2 ≤ k ≤ 8). Second, for the run with 32.6% B and 30 ◦ C, small changes in %B can result in a sizable decrease in resolution—which is less true for T = 30 ◦ Cand %B = 38%, so the latter separation is expected to be more robust. All of the latter considerations are immediately apparent from an inspection of Figure 10.3b. The user can carry out further simulations, such as a change in column condi- tions, without a need for additional laboratory experiments. Note that the pressure (2131 psi) is displayed, which can be used to avoid conditions that would overpres- sure the pump. As the resolution of this separation might be unacceptable (R s = 1.3), we can improve resolution by either increasing column length or decreasing particle size. Figure 10.3c shows the prediction for a change to a column of the same size (250 × 4.6-mm) packed with 3-μm particles, in place of the previous 5-μm column. Because of the increase in pressure due to the smaller particles, it is necessary to reduce the flow rate from 2.0 mL/min in Figure 10.3b to 1.0 mL/min in the separa- tion of Figure 10.3c. The resulting resolution (R s = 2.1) and pressure (P = 2960 psi) are acceptable, although the run time is doubled to 24 minutes. After exploring the options above, the best-possible separation might be considered inadequate because of an excessive run time. In this case a different column can be selected (or methanol can be substituted for acetonitrile as B-solvent), and the entire procedure repeated (four more experimental runs varying gradient time and temperature). 10.2.2 Gradient Optimization The four experimental runs of Figure 9.18 correspond to the experimental design of Figure 10.2b (varying temperature and gradient time). In this case, however, we will seek an optimized gradient separation (rather than an isocratic separation, as in Fig. 10.3). The resulting resolution map is shown in Figure 10.4a. The cross-hairs (and arrow) in the latter resolution map correspond to conditions for maximum resolution (R s = 2.1): a gradient of 5–100% in 35 minutes, and a temperature of 47 ◦ C. The resulting chromatogram is shown in Figure 10.4b. Because the last peak leaves the column at 14 minutes (before the gradient ends), the gradient can be shortened to 5–43%B in 14 minutes (see Section 9.3.4). Previously the (manual) trial-and-error optimization of gradient time and temperature for this sample yielded ‘‘optimized’’ conditions of 5–46%B gradient in 14 minutes at 50 ◦ C temperature, with a resulting resolution of R s = 1.9. In this example (Fig. 10.4) computer simulation provides only a slight improvement in resolution compared to manual, trial-and-error method development. However, computer simulation does confirm that the final conditions selected are truly ‘‘best,’’ since the resolution maps summarize predictions for several hundred combinations of conditions (different ‘‘virtual experiments’’). 484 COMPUTER-ASSISTED METHOD DEVELOPMENT 50 45 40 35 30 °C 80 70605040302010 Gradient time t G (min) 1.9 1.6 1.3 1.1 0.8 0.5 0.3 0.0 R s 02468101214 Time (min) 5−100% B in 35 min; 47°C R s = 2.1 1 2 3 4 5 − 7 8 9 10 11 2.2 (a) (b) Figure 10.4 Computer simulation for the optimization of gradient time and temperature. Same conditions and ‘irregular’ sample as for Figure 9.18 (5–100% B calibration gradients). (a) Resolution map; (b) optimized separation. A further inspection of the map of Figure 10.4a shows that resolution decreases only slightly for a shorter gradient time, especially if temperature is decreased at the same time. Thus, for a 5–46%B gradient in 12 min at 45 ◦ C, resolution is hardly changed (R s = 2.0), while run time is shortened by 2 minutes. Computer simulation is thus useful for fine-tuning a separation and so achieves maximum resolution and/or minimum run time by exploring further changes in conditions by trial and error. Minimizing run time in this fashion can be important when a method will be used for hundreds or thousands of samples. Method robustness can also be checked by making small changes in experimental conditions. In the example of Figure 10.4a a decrease in temperature by 2 ◦ C would result in a serious loss of resolution (R s = 1.3). When a method is carried out with a different HPLC system, uncertainties in the actual temperature of ±1–2 ◦ C are not uncommon (Section 3.7). The sensitivity of the present method to unintended changes in temperature can be greatly reduced by small changes in the final values of %B and T for the method, with only a modest sacrifice in resolution. For example, the choice of a gradient time of 13 minutes (5–46% B gradient) and a temperature of 48 ◦ C gives a resolution of R s = 1.9, but a change in temperature of ±2 ◦ C results in no loss of resolution; that is, the latter method is robust with respect to moderate temperature variations. Trial-and-error simulations of this kind can be carried out within a few seconds each (no new experiments required). 10.2 COMPUTER-SIMULATION SOFTWARE 485 100 80 60 40 20 0 %B 01234 5 678 9 (min) 5/30/100%B at 0/8.75/9.50 min T = 48°C;R s = 2.0 Figure 10.5 Trial-and-error computer simulations for the design of a segmented gradient (for an improved separation of Figure 10.4b). A large number of segmented gradients can be explored intuitively in a very short time by means of computer simulation, allowing the easy development of a final method that meets the needs of the user. This is illustrated in Figure 10.5, which shows the use of the DryLab computer screen for the design of such gradient shapes. The preceding example (t G = 13 min and 48 ◦ C) is the starting point. During the design of a gradient, each of the three inserted points shown in Figure 10.5 (marked by arrows) can be dragged via a computer mouse to any desired values of time and %B (thereby changing the gradient program), while the resulting chromatogram is simultaneously displayed as in Figure 10.5. The gradient shown in Figure 10.5 shortens the run time to 9.5 minutes, while maintaining a rugged separation with regard to small changes in temperature (see the related discussion of Section 9.2.2.5). 10.2.3 Other Features Computer-simulation software can offer the user a number of other options, as summarized in Table 10.1 (see items 4–9). 10.2.3.1 Isocratic Predictions from Gradient Data Once computer simulation has been initiated on the basis of two gradient exper- iments, where only gradient time is varied, it is also possible to predict isocratic separation as a function of %B. Similarly, when four experiments, as in Figure 10.2b, are carried out, isocratic separation as a function of %B and temperature can be predicted, as illustrated in Figure 10.3. This ability to predict isocratic separations on the basis of gradient experiments allows an evaluation of isocratic elution as a possible alternative to gradient separation. As the range in k for the sample decreases, isocratic elution becomes increasingly preferred. Predictions of isocratic separation from gradient data are somewhat less reliable, compared to the use of isocratic experiments for this purpose. Thus, if an initial gradient experiment suggests that isocratic elution is preferable (Section 9.3.1), the latter experiment might be followed by the four isocratic experiments of Figure 10.2a, rather than continuing with gradient experiments as in Figure 10.2b. (For a further discussion of the prediction of isocratic elution from gradient runs, see [14]). It should be . are usually easy to separate, the potential value of computer simulation may be at best marginal. Another barrier to the use of computer simulation is the belief that the chromatographer has no. predicted chromatograms are ‘‘ideal’’ rather than ‘‘real.’’ Thus baselines are assumed not to drift, peaks may be assumed to be symmetrical (although peak tailing can be taken into account by computer. column. Because of the increase in pressure due to the smaller particles, it is necessary to reduce the flow rate from 2.0 mL/min in Figure 10.3b to 1.0 mL/min in the separa- tion of Figure 10.3c.